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Question Number 142723 Answers: 1 Comments: 0

....... discrete ..... mathematics....... prove that: Σ_(n=1) ^∞ (1/(F_(2n+1) −1))=^? ((5−(√5))/2) F_n :: fibonacci sequence...

$$\:\:\:\:\:\:\:\:.......\:{discrete}\:\:.....\:\:{mathematics}....... \\ $$$$\:\:\:\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{F}_{\mathrm{2}{n}+\mathrm{1}} −\mathrm{1}}\overset{?} {=}\frac{\mathrm{5}−\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:{F}_{{n}} \:::\:{fibonacci}\:\:{sequence}... \\ $$

Question Number 142721 Answers: 1 Comments: 0

lim_(x→0) (x/( (√(1−cos x))))=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}}{\:\sqrt{\mathrm{1}−\mathrm{cos}\:{x}}}=? \\ $$

Question Number 142719 Answers: 0 Comments: 1

find the particular solution to the differential equation y^((4)) +21y^((2)) −100y=4(8−29t)e^(−2t) . solution please.

$$\mathrm{find}\:\mathrm{the}\:\mathrm{particular}\:\mathrm{solution} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\mathrm{y}^{\left(\mathrm{4}\right)} +\mathrm{21y}^{\left(\mathrm{2}\right)} −\mathrm{100y}=\mathrm{4}\left(\mathrm{8}−\mathrm{29t}\right)\mathrm{e}^{−\mathrm{2t}} . \\ $$$$\mathrm{solution}\:\mathrm{please}. \\ $$

Question Number 142717 Answers: 2 Comments: 1

Question Number 142765 Answers: 1 Comments: 1

A class has 13 children. To play a game one child is the referee and the other children are divided in three teams with four children in each team. In how many ways can the class play the game?

$${A}\:{class}\:{has}\:\mathrm{13}\:{children}.\:{To}\:{play}\:{a} \\ $$$${game}\:{one}\:{child}\:{is}\:{the}\:{referee}\:{and}\:{the} \\ $$$${other}\:{children}\:{are}\:{divided}\:{in}\:{three}\: \\ $$$${teams}\:{with}\:{four}\:{children}\:{in}\:{each}\:{team}. \\ $$$${In}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{class}\:{play} \\ $$$${the}\:{game}? \\ $$

Question Number 142763 Answers: 2 Comments: 2

Question Number 142712 Answers: 0 Comments: 0

Question Number 142710 Answers: 1 Comments: 0

Let k be non-negative real numbers and n ∈ N^+ ≥1. Prove that (((4−k)/(4+k)))(((12−k)/(12+k)))...[((2n(n+1)−k)/(2n(n+1)+k))] ≤ ((n+k+1)/((n+1)(k+1)))

$$\mathrm{Let}\:{k}\:\mathrm{be}\:\mathrm{non}-\mathrm{negative}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{and}\:{n}\:\in\:\mathrm{N}^{+} \geqslant\mathrm{1}.\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{4}−{k}}{\mathrm{4}+{k}}\right)\left(\frac{\mathrm{12}−{k}}{\mathrm{12}+{k}}\right)...\left[\frac{\mathrm{2}{n}\left({n}+\mathrm{1}\right)−{k}}{\mathrm{2}{n}\left({n}+\mathrm{1}\right)+{k}}\right]\:\leqslant\:\frac{{n}+{k}+\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({k}+\mathrm{1}\right)} \\ $$$$ \\ $$

Question Number 142708 Answers: 1 Comments: 0

I=∫_0 ^( α) (√(c^2 −sin^2 θ))dθ tan α=(a^2 /b^2 ) , a^2 >b^2 , c^2 >1 Perimeter of ellipse =4∫_0 ^( π/2) (√(a^2 −(a^2 −b^2 )sin^2 θ)) dθ (is that right sir?)

$$\:{I}=\int_{\mathrm{0}} ^{\:\:\alpha} \sqrt{{c}^{\mathrm{2}} −\mathrm{sin}\:^{\mathrm{2}} \theta}{d}\theta \\ $$$$\:\mathrm{tan}\:\alpha=\frac{{a}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:\:,\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \:\:,\:{c}^{\mathrm{2}} >\mathrm{1} \\ $$$${Perimeter}\:{of}\:{ellipse} \\ $$$$=\mathrm{4}\int_{\mathrm{0}} ^{\:\:\pi/\mathrm{2}} \sqrt{{a}^{\mathrm{2}} −\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)\mathrm{sin}\:^{\mathrm{2}} \theta}\:{d}\theta \\ $$$$\left({is}\:{that}\:{right}\:{sir}?\right) \\ $$

Question Number 142698 Answers: 1 Comments: 0

Prove that ∀n∈N^∗ Π_(k=1) ^(n−1) sin(((kπ)/(2n))) = Π_(k=1) ^(n−1) cos(((kπ)/(2n)))

$$\mathrm{Prove}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\prod}}\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{2n}}\right)\:=\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\prod}}\mathrm{cos}\left(\frac{\mathrm{k}\pi}{\mathrm{2n}}\right) \\ $$

Question Number 142691 Answers: 1 Comments: 1

The number of distributions of 52 cards divided equally to 4 persons so as each gets 4 cards of same suit taken away from 3suits(4×3=12)℘ remaining card from remaining 4 th suit is

$${The}\:{number}\:{of}\:{distributions}\:{of}\:\mathrm{52} \\ $$$${cards}\:{divided}\:{equally}\:{to}\:\mathrm{4}\:{persons}\:{so} \\ $$$${as}\:{each}\:{gets}\:\mathrm{4}\:{cards}\:{of}\:{same}\:{suit} \\ $$$${taken}\:{away}\:{from}\:\mathrm{3}{suits}\left(\mathrm{4}×\mathrm{3}=\mathrm{12}\right)\wp \\ $$$${remaining}\:{card}\:{from}\:{remaining} \\ $$$$\mathrm{4}\:{th}\:{suit}\:{is} \\ $$

Question Number 142690 Answers: 1 Comments: 0

∫_(0 ) ^1 ((log(x)log((x/(1−x))))/( (√(x/(1−x)))))dx Any help

$$\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{\boldsymbol{{log}}\left(\boldsymbol{{x}}\right)\boldsymbol{{log}}\left(\frac{\boldsymbol{{x}}}{\mathrm{1}−\boldsymbol{{x}}}\right)}{\:\sqrt{\frac{\boldsymbol{{x}}}{\mathrm{1}−\boldsymbol{{x}}}}}\boldsymbol{{dx}} \\ $$$$\boldsymbol{\mathrm{Any}}\:\boldsymbol{\mathrm{help}}\: \\ $$$$ \\ $$

Question Number 142689 Answers: 1 Comments: 0